[Paper Review] A tree-based model for addressing sparsity and taxa covariance in microbiome compositional count data

Zhuoqun Wang, Jialiang Mao+1 more|arXiv (Cornell University)|Jun 29, 2021

Geochemistry and Geologic Mapping21 references4 citations

TL;DR

This paper proposes the logistic-tree normal (LTN) model, a novel Bayesian generative model for microbiome compositional data that combines the flexible covariance structure of log-ratio normal (LN) models with the computational efficiency of Dirichlet-tree (DT) models via a tree-based binomial decomposition and Pólya-Gamma augmentation. The method enables scalable, high-dimensional inference with sparsity and low-rank assumptions, demonstrating strong performance in association testing and covariance estimation on a longitudinal T1D cohort dataset.

ABSTRACT

Microbiome compositional data are often high-dimensional, sparse, and exhibit pervasive cross-sample heterogeneity. Generative modeling is a popular approach to analyze such data, and effective generative models must accurately characterize these key features. While high-dimensionality and abundance of zeros have received much attention, existing models often lack flexibility in capturing complex cross-sample variability. This limitation can affect statistical efficiency and lead to misleading conclusions in tasks like differential abundance analysis, clustering, and network analysis. We introduce a generative model, the "logistic-tree normal" (LTN) model, which addresses this issue and effectively captures key characteristics of microbiome data, including abundance of zeros. LTN employs a tree-based decomposition to aggregate sparse taxa counts and uses a (multivariate) logistic-normal distribution at tree splits, allowing for flexible covariance adjustments among taxa as needed. The latent Gaussian structure of LTN enables the incorporation of multivariate analysis tools that enforce sparsity or low-rank covariance assumptions. As a versatile, fully generative model, LTN supports a wide range of applications and offers efficient Bayesian inference computational recipes through conjugate blocked Gibbs sampling with Pólya-Gamma augmentation. We demonstrate application of LTN in a compositional mixed-effects model for differential abundance analysis using numerical experiments and a reanalysis of the infant cohort in the DIABIMMUNE study. Our findings illustrate that LTN, by adequately accounting for cross-sample heterogeneity, appropriately generates the proportion of zeros without requiring an explicit zero-inflation component, confirming a recent viewpoint that "zero-inflation" in count-based sequencing data are often results of unaccounted cross-sample variation.

Motivation & Objective

• To address the limitations of existing models in handling complex covariance and computational scalability in high-dimensional microbiome compositional data.
• To develop a generative model that retains the rich covariance structure of log-ratio normal (LN) models while achieving computational tractability through tree-based decomposition.
• To enable effective Bayesian inference for microbiome association studies and covariance estimation using sparsity and low-rank assumptions.
• To demonstrate the utility of the LTN model in longitudinal microbiome data analysis, particularly in detecting associations with disease risk.

Proposed method

• The LTN model decomposes the multinomial likelihood into a series of binomial probabilities at the internal nodes of a phylogenetic tree.
• It models the log-odds of these binomial probabilities using a multivariate normal distribution, enabling flexible covariance structures among taxa.
• Pólya-Gamma auxiliary variables are introduced to enable efficient Gibbs sampling by leveraging conjugacy in the hierarchical model.
• The model supports sparsity and low-rank assumptions on the covariance matrix of log-odds, facilitating high-dimensional inference.
• A general mixed-effects model is constructed using LTN to model compositional random effects and test associations with covariates.
• The framework allows integration of prior information such as graphical Lasso on the inverse covariance matrix to infer sparse interaction networks among taxa.

Experimental results

Research questions

• RQ1 Can a model combine the flexible covariance of log-ratio normal models with the computational efficiency of tree-based models for microbiome data?
• RQ2 How can sparsity and low-rank structures be effectively incorporated into compositional data models to improve scalability and interpretability?
• RQ3 Can the LTN model detect meaningful associations between microbiome composition and disease risk in longitudinal studies?
• RQ4 How well does the LTN model estimate the underlying covariance structure among microbial taxa compared to existing methods?

Key findings

• The LTN model successfully captures complex covariance structures among microbial taxa, outperforming traditional Dirichlet-multinomial models in flexibility.
• The use of Pólya-Gamma augmentation enables efficient Gibbs sampling, making Bayesian inference scalable even with over 50 taxa.
• In the DIABIMMUNE T1D cohort study, the model detected significant associations between microbiome composition and dietary factors such as breastfeeding, solid food, and soy products.
• PMAPs revealed that dietary introduction of barley, rye, and solid food was linked to changes in the Firmicutes/Bacteroidetes ratio, a known dysbiosis indicator.
• The model identified a chain of nodes with consistent relative abundance changes for soy products, suggesting a cumulative effect on specific taxa like OTU 4439360.
• The analysis confirmed known biological patterns, such as Bifidobacterium enrichment during breastfeeding, particularly species like longum and bifidum.